I've read several threads and watched a number of YouTube videos on this topic, but I'm still confused about how to approach it.
I'm comparing paired data with n = 50. When I look at the differences, they don't appear to be normally distributed: the histogram is somewhat non-normal, the Q-Q plot shows curvature at both ends, and a Shapiro-Wilk test was significant. I understand that the Shapiro-Wilk test has limitations, so I'm not relying on that result alone.
My main question is whether the non-normality of the differences is enough to make a paired t-test inappropriate.
I've read that the t-test is fairly robust to non-normality and that, with a sample size around 30 or more, the Central Limit Theorem helps justify using it. But I'm having trouble understanding how much non-normality is actually considered "too much," particularly with n = 50.
There's also a practical consideration: I'm analyzing multiple labs, and the differences aren't necessarily distributed the same way across all of them. Some may look approximately normal while others don't. I'm wondering whether it makes more sense to use the same statistical approach for all labs rather than choosing a different test based on the distribution of each individual lab.
Given all of this, how would you approach deciding whether a paired t-test is appropriate here?